The minimum value of x which satisfies the inequality (si n ^ - 1...

The minimum value of

x

which satisfies the inequality ${\left(si{n}^{- 1}x\right)}^{2} \ge {\left(co{s}^{- 1}x\right)}^{2} is The minimum value of x which satisfies the inequality (si n ^ - 1 x) ^ 2 ≥ (co s ^ - 1 x) ^ 2 is

Options

A $\dfrac{1}{\sqrt{2}}$
B $\dfrac{1}{2}$
C $\dfrac{\sqrt{3}}{2}$
D $\dfrac{1}{\sqrt{3}}$

Correct answer

A. $\dfrac{1}{\sqrt{2}}$

Step-by-step solution

{\left(si{n}^{- 1}x\right)}^{2} - {\left(co{s}^{- 1}x\right)}^{2} \ge 0

\left(si{n}^{- 1}x + co{s}^{- 1}x\right)\left(si{n}^{- 1}x - co{s}^{- 1}x\right) \ge 0

<br />

⇒ si{n}^{- 1}x \ge co{s}^{- 1}x

<br />Clearly from graph<br />

x \in \left[\dfrac{1}{\sqrt{2}}, 1\right]

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